Correlation, causation, and coupled pairs of differential equations

An aspect of paleoclimate evidence to which Professor Jennifer Francis alludes in her recent report on Arctic amplification is the close mutual modeling which Earth surface temperature and carbon dioxide concentration exhibit during the recent geologic past. Since relative timings are hard to discern in these series, due to imprecision of dating, this has sometimes been seized by some people, ignorant of physics, to argue that changes in carbon dioxide concentrations may be a consequence of temperature changes, and not their cause. Indeed, they are closely linked.

So close, in fact, that their relative values are bound up in a system of coupled differential equations. In these systems, behavior is interdependent so the value of one physical variable is determined by the other.  If, in our case, carbon dioxide were to increase dramatically, the response of the coupled partner is to similarly increase, meaning temperature increase. And, on the other side, if temperature increases, more CO2 is released as well.

Such coupled pairs of quantities aren’t unusual, although having  two which mutually re-enforce one another is slightly unusual.  Typically one quantity is an antagonist to the other, and these are part, therefore, of a feedback control system.

In the case of temperature, it is contained, ultimately, by radiation to space, via the Blackbody Effect.  In the case of CO2, it is scrubbed from atmosphere primarily by being dissolved in ocean and, there, by being incorporated into carbonate shells by living things and as biological detritus, and secondarily as plant biomass on land.  These are slow processes, although significant at geological time scales.

A text which shows how useful and pervasive such systems are is S. P. Ellner, J. Guckenheimer, Dynamic Models in Biology, Princeton University Press, 2006.

Afterthought: Here’s a way of looking at this which might be conceptually easier. Suppose there’s this circular conveyor belt, and Fred and Jane are standing on opposite sides of it. On the belt are a number of small blocks, some red, some green. When each red block passes Jane, she adds another green block to the belt. (She has a very large number of green blocks available in a huge bag.) When each green block passes Fred, he adds two red blocks to the belt.  (He also has a huge bag.) A quarter way ’round the circle stands Bob.  He has a large shovel, and he periodically removes 1000 blocks at a time from the belt, or as many as he has in front of him. The belt is moving slowly enough for Fred, Jane, and Bob to fulfill their roles.

What happens depends in part on how many blocks there are on the belt initially, and partly their colors. If there are many green blocks initially, it’s more likely Bob won’t be able to keep up with the growing number of blocks, since he only removes a certain number per unit time.

Here’s the question … Does the presence of green blocks cause the increase in the number of red blocks?  Or does the presence of red blocks cause the increase in the number of green blocks?  It’s only sensible to think neither or both.

Now, suppose Harry walks in with a huge bag of green blocks.  Instead of putting these on the belt, he just dumps them onto the belt.  What happens to the number of red and green blocks?

What wants to be set up here is a Lotka-Volterra pair of quantities, e.g., like that described here.  I’ll do this better some time in the future.

So, in case the metaphor is too thick, this is an analogy to carbon dioxide, the green blocks, and temperature, the red blocks. And humanity is in the role of Harry.


About hypergeometric

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This entry was posted in chemistry, climate, climate education, ecology, economics, engineering, environment, geoengineering, geophysics, mathematics, meteorology, oceanography, physics, rationality, reasonableness, statistics, stochastic algorithms. Bookmark the permalink.

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